Finite Group Databasebeta

← #13089414

Subgroups — #13089414

Order 1440.

696 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3830C211yesmaximal, characteristic, complemented
2C3×#6819C211yesmaximal, characteristic, complemented
3C3×D4×D15C211yesmaximal, characteristic, complemented
4#10624235C212yesmaximal, complemented
5#10624230C212yesmaximal, complemented
6#10624229C212yesmaximal, complemented
7C2×#67154C212yesmaximal, complemented
8C4×(C3⋊D3)⋊D5C211yesmaximal, characteristic, complemented
9#10624221C211yesmaximal, characteristic, complemented
10#10624220C211yesmaximal, characteristic, complemented
11#10624219C211yesmaximal, characteristic, complemented
12#92024—31nomaximal
13D5×#237—31nomaximal
14C22×C3×D15C2212yes—
15C2×(C3⋊D3)⋊D5C2211yescharacteristic
16C32×C5×D4C2211yescharacteristic, complemented
17C5×#638C2212yescomplemented
18C2×C5×C3⋊Dic3C2212yescomplemented
19C4×C5×C3⋊D3C2211yescharacteristic, complemented
20C5×#634C2211yescharacteristic, complemented
21C3×#1220C2212yescomplemented
22C3×#1220C2212yescomplemented
23C2×C3×Dic15C2212yescomplemented
24C3×D60C2211yescharacteristic, complemented
25C3×C4×D15C2211yescharacteristic
26C3×C4×D15C2211yescharacteristic, complemented
27C3×Dic30C2211yescharacteristic, complemented
28#67158C2212yescomplemented
29#67157C2212yescomplemented
30#67156C2212yescomplemented
31#67155C2211yescharacteristic, complemented
32#67154C2212yes—
33#67154C2211yescharacteristic
34#67154C2212yescomplemented
35#67154C2212yescomplemented
36#67154C2211yescharacteristic, complemented
37D3×#237—51nomaximal, Hall
38#6819D311yescharacteristic, complemented
39D4×D15D311yescharacteristic, complemented
40C5×#237—31no—
41C5×#237—31no—
42C5×#237—61no—
43C3×#711—31no—
44C3×D4×D5—31no—
45#6790—32no—
46D5×#67—32no—
47C2×#1200—32no—
48#6782—32no—
49#6780—32no—
50#6779—32no—
51#6779—32no—
52C2×D5×Dic3—32no—
53C4×D3×D5—31no—
54C4×D3×D5—31no—
55#6773—31no—
56D15⋊Q8—31no—
57#6769—31no—
58#6766—31no—
59#6765—31no—
60D5×Dic6—31no—
61C22×C32×C5C2312yes—
62C2×C5×C3⋊D3C2311yescharacteristic
63C2×C3×D15—24no—
64C2×C3×D15C2312yes—
65C2×C3×D15C2311yescharacteristic
66C2×C3×D15C2311yescharacteristic
67(C3⋊D3)⋊D5—22no—
68C32×C4×C5C2311yescharacteristic
69C5×C3⋊Dic3C2311yescharacteristic
70C5×C3⋊Dic3C2312yes—
71C3×Dic15C2311yescharacteristic
72C3×Dic15C2312yes—
73C3×Dic15C2311yescharacteristic
74D5×#41—91noHall
75#3830D511yescharacteristic, complemented
76C3×#237—51no—
77C3×D3×D4—51no—
78D3×#67—52no—
79#3805—52no—
80#3804—52no—
81C2×D3×Dic3—52no—
82C4×D32—51no—
83#3796—51no—
84#3795—51no—
85D3×Dic6—51no—
86C22×D15C2×D312yes—
87C22×C3×D5—32no—
88C2×D3×D5—31no—
89C2×D3×D5—31no—
90C3×C5×D4—21no—
91C3×C5×D4C2×D311yescharacteristic, complemented
92C3×C5×D4C2×D311yescharacteristic, complemented
93#1220C2×D312yescomplemented
94#1220C2×D312yescomplemented
95C2×Dic15C2×D312yescomplemented
96D60C2×D311yescharacteristic, complemented
97C4×D15C2×D311yescharacteristic
98C4×D15C2×D311yescharacteristic, complemented
99Dic30C2×D311yescharacteristic, complemented
100C5×#67—32no—
101C5×#67—32no—
102C5×#67—62no—
103C2×C5×Dic3—32no—
104C2×C5×Dic3—32no—
105C2×C5×Dic3—62no—
106C4×C5×D3—31no—
107C4×C5×D3—31no—
108C4×C5×D3—61no—
109C5×Dic6—31no—
110C5×Dic6—31no—
111C5×Dic6—61no—
112C3×C22⋊D5—32no—
113C3×C22⋊D5—32no—
114C2×C3×Dic5—32no—
115C3×D20—31no—
116C3×C4×D5—31no—
117C3×C4×D5—31no—
118C3×Dic10—31no—
119#1204—32no—
120#1204—32no—
121#1203—32no—
122#1202—32no—
123#1202—32no—
124#1201—32no—
125#1200—32no—
126#1200—31no—
127#1200—32no—
128#1200—32no—
129#1200—31no—
130D3×Dic5—31no—
131D3×Dic5—31no—
132D5×Dic3—32no—
133D5×Dic3—31no—
134D5×Dic3—32no—
135D5×Dic3—32no—
136D5×Dic3—31no—
137D3×#41—151no—
138C2×#237—151no—
139C2×C32×C5—22no—
140C2×C32×C5C2411yescharacteristic
141C5×C3⋊D3—21no—
142C3×D15—21no—
143C3×D15—22no—
144C3×D15#4112yescomplemented
145C5×#41—91no—
146#713—91no—
147Q8×D5—91no—
148#711—31no—
149#711—92no—
150D4×D5—31no—
151D4×D5—92no—
152#709—91no—
153#709—92no—
154C2×C4×D5—91no—
155C2×C4×D5—92no—
156C22×C3×D3—52no—
157C2×D32—51no—
158C32×D4C2×D511yescharacteristic, complemented
159#638C2×D512yescomplemented
160C2×C3⋊Dic3C2×D512yescomplemented
161C4×C3⋊D3C2×D511yescharacteristic, complemented
162#634C2×D511yescharacteristic, complemented
163C3×#67—52no—
164C3×#67—52no—
165C2×C3×Dic3—52no—
166C3×D12—51no—
167C3×C4×D3—51no—
168C3×C4×D3—51no—
169C3×Dic6—51no—
170#627—52no—
171#626—52no—
172#625—52no—
173#624—51no—
174D3×Dic3—52no—
175D3×Dic3—51no—
176D3×Dic3—52no—
177D3×Dic3—52no—
178D3×Dic3—51no—
179C22×C3×C5—22no—
180C22×C3×C5C22×D312yes—
181C22×C3×C5C22×D312yes—
182C2×D15—24no—
183C2×D15C22×D312yes—
184C2×D15C22×D311yescharacteristic
185C2×D15C22×D311yescharacteristic
186C2×C5×D3—31no—
187C2×C5×D3—31no—
188C2×C5×D3—61no—
189C2×C3×D5—31no—
190C2×C3×D5—32no—
191C2×C3×D5—31no—
192C2×C3×D5—64no—
193D3×D5—62no—
194D3×D5—62no—
195C3×C4×C5—21no—
196C3×C4×C5C22×D311yescharacteristic
197C3×C4×C5C22×D311yescharacteristic
198Dic15C22×D311yescharacteristic
199Dic15C22×D312yes—
200Dic15C22×D311yescharacteristic
201C3×Dic5—31no—
202C3×Dic5—31no—
203C3×Dic5—32no—
204C5×Dic3—31no—
205C5×Dic3—32no—
206C5×Dic3—31no—
207C5×Dic3—32no—
208C5×Dic3—62no—
209C5×Dic3—61no—
210C3×#41—151no—
211C2×C3×D4—151no—
212C2×#67—152no—
213C22×Dic3—152no—
214#239—151no—
215D3×Q8—151no—
216#237—31no—
217#237—31no—
218#237—51no—
219#237—61no—
220#237—151no—
221#237—151no—
222#237—152no—
223#237—152no—
224#237—152no—
225D3×D4—51no—
226D3×D4—152no—
227#235—151no—
228#235—152no—
229C2×C4×D3—151no—
230C2×C4×D3—151no—
231C2×C4×D3—152no—
232C2×Dic6—151no—
233C32×C5C2×#4111yescharacteristic, complemented, Hall
234C22×D5—32no—
235C22×D5—91no—
236C5×Q8—91no—
237C5×D4—92no—
238C5×D4D3211yescharacteristic, complemented
239C2×C4×C5—92no—
240C2×C4×C5—91no—
241C22⋊D5—32no—
242C22⋊D5—32no—
243C22⋊D5—92no—
244C2×Dic5—32no—
245C2×Dic5—91no—
246D20—31no—
247D20—92no—
248C4×D5—31no—
249C4×D5—31no—
250C4×D5—91no—
251C4×D5—92no—
252C4×D5—91no—
253C4×D5—92no—
254C4×D5—92no—
255Dic10—31no—
256Dic10—92no—
257C22×C32C22×D512yes—
258C2×C3⋊D3C22×D511yescharacteristic
259C2×C3×D3—51no—
260C2×C3×D3—52no—
261C2×C3×D3—51no—
262C2×C3×D3—104no—
263D32—102no—
264C32×C4C22×D511yescharacteristic
265C3⋊Dic3C22×D511yescharacteristic
266C3⋊Dic3C22×D512yes—
267C3×Dic3—51no—
268C3×Dic3—52no—
269C3×Dic3—51no—
270C2×#41—451noSylow
271C2×C3×C5—22no—
272C2×C3×C5—22no—
273C2×C3×C5—21no—
274C2×C3×C5—42no—
275C2×C3×C5C23×D311yescharacteristic
276C2×C3×C5C23×D311yescharacteristic
277D15—22no—
278D15—21no—
279D15#23712yescomplemented
280C3×D5—32no—
281C3×D5—61no—
282C3×D5—62no—
283C5×D3—61no—
284C5×D3—61no—
285C5×D3—121no—
286C23×C3—152no—
287C22×D3—52no—
288C22×D3—151no—
289C22×D3—151no—
290C3×Q8—151no—
291C3×D4—21no—
292C3×D4—152no—
293C3×D4—152no—
294C3×D4—151no—
295C3×D4D3×D511yescharacteristic, complemented
296C3×D4D3×D511yescharacteristic, complemented
297C2×C3×C4—152no—
298C2×C3×C4—151no—
299C2×C3×C4—151no—
300#67—32no—
301#67—32no—
302#67—52no—
303#67—52no—
304#67—62no—
305#67—152no—
306#67—152no—
307#67—152no—
308C2×Dic3—32no—
309C2×Dic3—32no—
310C2×Dic3—52no—
311C2×Dic3—62no—
312C2×Dic3—152no—
313C2×Dic3—151no—
314C2×Dic3—151no—
315C2×Dic3—152no—
316C2×Dic3—152no—
317D12—51no—
318D12—152no—
319C4×D3—31no—
320C4×D3—31no—
321C4×D3—51no—
322C4×D3—51no—
323C4×D3—61no—
324C4×D3—151no—
325C4×D3—152no—
326C4×D3—152no—
327C4×D3—151no—
328C4×D3—152no—
329C4×D3—151no—
330C4×D3—151no—
331Dic6—31no—
332Dic6—31no—
333Dic6—51no—
334Dic6—61no—
335Dic6—152no—
336Dic6—152no—
337C22×C5—91no—
338C22×C5C2×D3212yes—
339C2×D5—31no—
340C2×D5—31no—
341C2×D5—32no—
342C2×D5—64no—
343C2×D5—182no—
344C4×C5—92no—
345C4×C5—91no—
346C4×C5C2×D3211yescharacteristic
347Dic5—32no—
348Dic5—31no—
349Dic5—31no—
350C2×C32—22no—
351C2×C32C23×D511yescharacteristic
352C3⋊D3—21no—
353C3×D3—52no—
354C3×D3—101no—
355C3×D3—102no—
356#41—91no—
357#41—151no—
358#41—452no—
359#41—451no—
360#41—452no—
361#41—451no—
362C2×Q8—451no—
363C2×D4—151no—
364C2×D4—452no—
365C22×C4—451no—
366C22×C4—452no—
367C3×C5—21no—
368C3×C5C2×#23711yescharacteristic, complemented
369C3×C5D3×#4111yescharacteristic, complemented
370C22×C3—22no—
371C22×C3—151no—
372C22×C3—152no—
373C22×C3—151no—
374C22×C3—304no—
375C22×C3C2×D3×D512yes—
376C22×C3C2×D3×D512yes—
377C2×D3—31no—
378C2×D3—31no—
379C2×D3—51no—
380C2×D3—51no—
381C2×D3—52no—
382C2×D3—61no—
383C2×D3—104no—
384C2×D3—302no—
385C2×D3—302no—
386C3×C4—21no—
387C3×C4—151no—
388C3×C4—152no—
389C3×C4—151no—
390C3×C4C2×D3×D511yescharacteristic
391C3×C4C2×D3×D511yescharacteristic
392Dic3—32no—
393Dic3—32no—
394Dic3—31no—
395Dic3—31no—
396Dic3—52no—
397Dic3—51no—
398Dic3—51no—
399Dic3—62no—
400Dic3—61no—
401C2×C5—22no—
402C2×C5—181no—
403C2×C5C22×D3211yescharacteristic
404D5—32no—
405D5—62no—
406D5—61no—
407C32D5×#4111yescharacteristic, complemented, Sylow
408C23—152no—
409C23—451no—
410Q8—91no—
411Q8—151no—
412Q8—452no—
413D4—92no—
414D4—152no—
415D4—152no—
416D4—151no—
417D4—452no—
418D4—452no—
419D4(C3⋊D3)⋊D511yescharacteristic, complemented
420C2×C4—92no—
421C2×C4—91no—
422C2×C4—152no—
423C2×C4—151no—
424C2×C4—151no—
425C2×C4—451no—
426C2×C4—452no—
427C2×C4—451no—
428C2×C4—452no—
429C2×C4—452no—
430C2×C4—451no—
431C2×C3—22no—
432C2×C3—22no—
433C2×C3—21no—
434C2×C3—42no—
435C2×C3—152no—
436C2×C3—301no—
437C2×C3—302no—
438C2×C3C22×D3×D511yescharacteristic
439C2×C3C22×D3×D511yescharacteristic
440D3—52no—
441D3—61no—
442D3—61no—
443D3—102no—
444D3—101no—
445D3—121no—
446C5D3×#23711yescharacteristic, complemented, Sylow
447C22—91no—
448C22—151no—
449C22—151no—
450C22—152no—
451C22—304no—
452C22—902no—
453C22C2×(C3⋊D3)⋊D512yes—
454C4—91no—
455C4—92no—
456C4—151no—
457C4—151no—
458C4—152no—
459C4C2×(C3⋊D3)⋊D511yescharacteristic
460C3—21no—
461C3D5×#23711yescharacteristic, complemented
462C3#9202411yescharacteristic, complemented
463C2—22no—
464C2—152no—
465C2—181no—
466C2—302no—
467C2—301no—
468C2C22×(C3⋊D3)⋊D511yescharacteristic
subgroupG/H
center, FrattiniC2C22×(C3⋊D3)⋊D5
commutator, socle, cosocleC2×C32×C5C24
FittingC32×C5×D4C22
hypercenterD4(C3⋊D3)⋊D5
hypocenterC32×C5C2×#41
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ D4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C5×#3830C2semidirect
2. C3×#6819C2semidirect
3. C3×D4×D15C2semidirect
4. #10624235C2semidirect
5. #10624230C2semidirect
6. #10624229C2semidirect
7. C2×#67154C2semidirect
8. C4×(C3⋊D3)⋊D5C2semidirect
9. #10624221C2semidirect
10. #10624220C2semidirect
11. #10624219C2semidirect
16. C32×C5×D4C22semidirect
17. C5×#638C22semidirect
18. C2×C5×C3⋊Dic3C22semidirect
19. C4×C5×C3⋊D3C22semidirect
20. C5×#634C22semidirect
21. C3×#1220C22semidirect
22. C3×#1220C22semidirect
23. C2×C3×Dic15C22semidirect
24. C3×D60C22semidirect
26. C3×C4×D15C22semidirect
27. C3×Dic30C22semidirect
28. #67158C22semidirect
29. #67157C22semidirect
30. #67156C22semidirect
31. #67155C22semidirect
34. #67154C22semidirect
35. #67154C22semidirect
36. #67154C22semidirect
38. #6819D3semidirect
39. D4×D15D3semidirect
75. #3830D5semidirect
91. C3×C5×D4C2×D3semidirect
92. C3×C5×D4C2×D3semidirect
93. #1220C2×D3semidirect
94. #1220C2×D3semidirect
95. C2×Dic15C2×D3semidirect
96. D60C2×D3semidirect
98. C4×D15C2×D3semidirect
99. Dic30C2×D3semidirect
144. C3×D15#41semidirect
158. C32×D4C2×D5semidirect
159. #638C2×D5semidirect
160. C2×C3⋊Dic3C2×D5semidirect
161. C4×C3⋊D3C2×D5semidirect
162. #634C2×D5semidirect
233. C32×C5C2×#41semidirect
238. C5×D4D32semidirect
279. D15#237semidirect
295. C3×D4D3×D5semidirect
296. C3×D4D3×D5semidirect
368. C3×C5C2×#237semidirect
369. C3×C5D3×#41semidirect
407. C32D5×#41semidirect
419. D4(C3⋊D3)⋊D5semidirect
446. C5D3×#237semidirect
461. C3D5×#237semidirect
462. C3#92024semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C5×#38302
C3×#68192
C3×D4×D152
#106242352
#106242302
#106242292
C2×#671542
C4×(C3⋊D3)⋊D52
#106242212
#106242202
#106242192
#920243
D5×#2373
D3×#2375